Questions — OCR (4619 questions)

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OCR C1 2009 January Q4
6 marks Moderate -0.8
4
  1. Sketch the curve \(y = \frac { 1 } { x ^ { 2 } }\).
  2. The curve \(y = \frac { 1 } { x ^ { 2 } }\) is translated by 3 units in the negative \(x\)-direction. State the equation of the curve after it has been translated.
  3. The curve \(y = \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor 4 and, as a result, the point \(P ( 1,1 )\) is transformed to the point \(Q\). State the coordinates of \(Q\).
OCR C1 2009 January Q5
9 marks Easy -1.3
5 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:
  1. \(y = 10 x ^ { - 5 }\),
  2. \(y = \sqrt [ 4 ] { x }\),
  3. \(y = x ( x + 3 ) ( 1 - 5 x )\).
OCR C1 2009 January Q6
8 marks Moderate -0.8
6
  1. Express \(5 x ^ { 2 } + 20 x - 8\) in the form \(p ( x + q ) ^ { 2 } + r\).
  2. State the equation of the line of symmetry of the curve \(y = 5 x ^ { 2 } + 20 x - 8\).
  3. Calculate the discriminant of \(5 x ^ { 2 } + 20 x - 8\).
  4. State the number of real roots of the equation \(5 x ^ { 2 } + 20 x - 8 = 0\).
OCR C1 2009 January Q7
8 marks Moderate -0.8
7 The line with equation \(3 x + 4 y - 10 = 0\) passes through point \(A ( 2,1 )\) and point \(B ( 10 , k )\).
  1. Find the value of \(k\).
  2. Calculate the length of \(A B\). A circle has equation \(( x - 6 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).
  3. Write down the coordinates of the centre and the radius of the circle.
  4. Verify that \(A B\) is a diameter of the circle.
OCR C1 2009 January Q8
10 marks Moderate -0.3
8
  1. Solve the equation \(5 - 8 x - x ^ { 2 } = 0\), giving your answers in simplified surd form.
  2. Solve the inequality \(5 - 8 x - x ^ { 2 } \leqslant 0\).
  3. Sketch the curve \(y = \left( 5 - 8 x - x ^ { 2 } \right) ( x + 4 )\), giving the coordinates of the points where the curve crosses the coordinate axes.
OCR C1 2009 January Q9
7 marks Moderate -0.3
9 The curve \(y = x ^ { 3 } + p x ^ { 2 } + 2\) has a stationary point when \(x = 4\). Find the value of the constant \(p\) and determine whether the stationary point is a maximum or minimum point.
OCR C1 2009 January Q10
12 marks Standard +0.3
10 A curve has equation \(y = x ^ { 2 } + x\).
  1. Find the gradient of the curve at the point for which \(x = 2\).
  2. Find the equation of the normal to the curve at the point for which \(x = 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  3. Find the values of \(k\) for which the line \(y = k x - 4\) is a tangent to the curve.
OCR C1 2010 January Q1
3 marks Easy -1.2
1 Express \(x ^ { 2 } - 12 x + 1\) in the form \(( x - p ) ^ { 2 } + q\).
OCR C1 2010 January Q2
4 marks Easy -1.2
2
\includegraphics[max width=\textwidth, alt={}, center]{918d83c3-1608-4482-9d3d-8af05e65f353-2_330_681_390_731} The graph of \(y = \mathrm { f } ( x )\) for \(- 2 \leqslant x \leqslant 4\) is shown above.
  1. Sketch the graph of \(y = 2 \mathrm { f } ( x )\) for \(- 2 \leqslant x \leqslant 4\) on the axes provided.
  2. Describe the transformation which transforms the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { f } ( x - 1 )\).
OCR C1 2010 January Q3
7 marks Moderate -0.8
3 Find the equation of the normal to the curve \(y = x ^ { 3 } - 4 x ^ { 2 } + 7\) at the point \(( 2 , - 1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 2010 January Q4
7 marks Easy -1.3
4 Solve the equations
  1. \(3 ^ { m } = 81\),
  2. \(\left( 36 p ^ { 4 } \right) ^ { \frac { 1 } { 2 } } = 24\),
  3. \(5 ^ { n } \times 5 ^ { n + 4 } = 25\).
OCR C1 2010 January Q5
7 marks Standard +0.3
5 Solve the equation \(x - 8 \sqrt { x } + 13 = 0\), giving your answers in the form \(p \pm q \sqrt { r }\), where \(p , q\) and \(r\) are integers.
OCR C1 2010 January Q6
7 marks Easy -1.3
6
\includegraphics[max width=\textwidth, alt={}, center]{918d83c3-1608-4482-9d3d-8af05e65f353-2_394_846_1868_648} The diagram shows part of the curve \(y = x ^ { 2 } + 5\). The point \(A\) has coordinates ( 1,6 ). The point \(B\) has coordinates ( \(a , a ^ { 2 } + 5\) ), where \(a\) is a constant greater than 1 . The point \(C\) is on the curve between \(A\) and \(B\).
  1. Find by differentiation the value of the gradient of the curve at the point \(A\).
  2. The line segment joining the points \(A\) and \(B\) has gradient 2.3. Find the value of \(a\).
  3. State a possible value for the gradient of the line segment joining the points \(A\) and \(C\).
OCR C1 2010 January Q7
5 marks Easy -1.3
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{918d83c3-1608-4482-9d3d-8af05e65f353-3_618_606_255_397} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{918d83c3-1608-4482-9d3d-8af05e65f353-3_622_622_251_1128} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{918d83c3-1608-4482-9d3d-8af05e65f353-3_620_613_986_395} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{918d83c3-1608-4482-9d3d-8af05e65f353-3_620_611_986_1128} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Each diagram shows a quadratic curve. State which diagram corresponds to the curve
    (a) \(y = ( 3 - x ) ^ { 2 }\),
    (b) \(y = x ^ { 2 } + 9\),
    (c) \(y = ( 3 - x ) ( x + 3 )\).
  2. Give the equation of the curve which does not correspond to any of the equations in part (i).
OCR C1 2010 January Q8
9 marks Moderate -0.3
8 A circle has equation \(x ^ { 2 } + y ^ { 2 } + 6 x - 4 y - 4 = 0\).
  1. Find the centre and radius of the circle.
  2. Find the coordinates of the points where the circle meets the line with equation \(y = 3 x + 4\).
OCR C1 2010 January Q9
8 marks Moderate -0.8
9 Given that \(\mathrm { f } ( x ) = \frac { 1 } { x } - \sqrt { x } + 3\),
  1. find \(\mathrm { f } ^ { \prime } ( x )\),
  2. find \(\mathrm { f } ^ { \prime \prime } ( 4 )\).
OCR C1 2010 January Q10
4 marks Moderate -0.3
10 The quadratic equation \(k x ^ { 2 } - 30 x + 25 k = 0\) has equal roots. Find the possible values of \(k\).
OCR C1 2010 January Q11
11 marks Standard +0.3
11 A lawn is to be made in the shape shown below. The units are metres.
\includegraphics[max width=\textwidth, alt={}, center]{918d83c3-1608-4482-9d3d-8af05e65f353-4_412_698_486_726}
  1. The perimeter of the lawn is \(P \mathrm {~m}\). Find \(P\) in terms of \(x\).
  2. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the lawn is given by \(A = 9 x ^ { 2 } + 6 x\). The perimeter of the lawn must be at least 39 m and the area of the lawn must be less than \(99 \mathrm {~m} ^ { 2 }\).
  3. By writing down and solving appropriate inequalities, determine the set of possible values of \(x\).
OCR C1 2011 January Q1
7 marks Moderate -0.8
1 The points \(A\) and \(B\) have coordinates \(( 6,1 )\) and \(( - 2,7 )\) respectively.
  1. Find the length of \(A B\).
  2. Find the gradient of the line \(A B\).
  3. Determine whether the line \(4 x - 3 y - 10 = 0\) is perpendicular to \(A B\).
OCR C1 2011 January Q2
3 marks Moderate -0.3
2 Given that $$( x - p ) \left( 2 x ^ { 2 } + 9 x + 10 \right) = \left( x ^ { 2 } - 4 \right) ( 2 x + q )$$ for all values of \(x\), find the constants \(p\) and \(q\).
OCR C1 2011 January Q3
5 marks Easy -1.8
3 Express each of the following in the form \(8 ^ { p }\) :
  1. \(\sqrt { 8 }\),
  2. \(\frac { 1 } { 64 }\),
  3. \(2 ^ { 6 } \times 2 ^ { 2 }\).
OCR C1 2011 January Q4
6 marks Moderate -0.5
4 By using the substitution \(u = ( 3 x - 2 ) ^ { 2 }\), find the roots of the equation $$( 3 x - 2 ) ^ { 4 } - 5 ( 3 x - 2 ) ^ { 2 } + 4 = 0$$
OCR C1 2011 January Q5
6 marks Easy -1.2
5
  1. Sketch the curve \(y = - x ^ { 3 }\).
  2. The curve \(y = - x ^ { 3 }\) is translated by 3 units in the positive \(x\)-direction. Find the equation of the curve after it has been translated.
  3. Describe a transformation that transforms the curve \(y = - x ^ { 3 }\) to the curve \(y = - 5 x ^ { 3 }\).
OCR C1 2011 January Q6
6 marks Moderate -0.8
6 Given that \(y = \frac { 5 } { x ^ { 2 } } - \frac { 1 } { 4 x } + x\), find
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
OCR C1 2011 January Q7
11 marks Moderate -0.3
7
  1. Express \(4 x ^ { 2 } + 12 x - 3\) in the form \(p ( x + q ) ^ { 2 } + r\).
  2. Solve the equation \(4 x ^ { 2 } + 12 x - 3 = 0\), giving your answers in simplified surd form.
  3. The quadratic equation \(4 x ^ { 2 } + 12 x - k = 0\) has equal roots. Find the value of \(k\).